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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcvat | Structured version Visualization version GIF version |
Description: A nonzero subspace less than the sum of two atoms is an atom. (atcvati 32414 analog.) (Contributed by NM, 10-Jan-2015.) |
Ref | Expression |
---|---|
lsatcvat.o | ⊢ 0 = (0g‘𝑊) |
lsatcvat.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcvat.p | ⊢ ⊕ = (LSSum‘𝑊) |
lsatcvat.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcvat.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcvat.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcvat.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
lsatcvat.r | ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
lsatcvat.n | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
lsatcvat.l | ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
Ref | Expression |
---|---|
lsatcvat | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcvat.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
2 | lsatcvat.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lsatcvat.p | . . 3 ⊢ ⊕ = (LSSum‘𝑊) | |
4 | lsatcvat.a | . . 3 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
5 | lsatcvat.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑊 ∈ LVec) |
7 | lsatcvat.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
9 | lsatcvat.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
11 | lsatcvat.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴) | |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
13 | lsatcvat.n | . . . 4 ⊢ (𝜑 → 𝑈 ≠ { 0 }) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
15 | lsatcvat.l | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ⊊ (𝑄 ⊕ 𝑅)) |
17 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → ¬ 𝑄 ⊆ 𝑈) | |
18 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17 | lsatcvatlem 39030 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑄 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
19 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑊 ∈ LVec) |
20 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝑆) |
21 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑅 ∈ 𝐴) |
22 | 9 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑄 ∈ 𝐴) |
23 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ≠ { 0 }) |
24 | lveclmod 21122 | . . . . . . . . 9 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
25 | 5, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod) |
26 | lmodabl 20923 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Abel) |
28 | 2 | lsssssubg 20973 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
29 | 25, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
30 | 2, 4, 25, 9 | lsatlssel 38978 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
31 | 29, 30 | sseldd 3995 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (SubGrp‘𝑊)) |
32 | 2, 4, 25, 11 | lsatlssel 38978 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
33 | 29, 32 | sseldd 3995 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (SubGrp‘𝑊)) |
34 | 3 | lsmcom 19890 | . . . . . . 7 ⊢ ((𝑊 ∈ Abel ∧ 𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊)) → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
35 | 27, 31, 33, 34 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝑄 ⊕ 𝑅) = (𝑅 ⊕ 𝑄)) |
36 | 35 | psseq2d 4105 | . . . . 5 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) ↔ 𝑈 ⊊ (𝑅 ⊕ 𝑄))) |
37 | 15, 36 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
38 | 37 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ⊊ (𝑅 ⊕ 𝑄)) |
39 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → ¬ 𝑅 ⊆ 𝑈) | |
40 | 1, 2, 3, 4, 19, 20, 21, 22, 23, 38, 39 | lsatcvatlem 39030 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑅 ⊆ 𝑈) → 𝑈 ∈ 𝐴) |
41 | 29, 7 | sseldd 3995 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
42 | 3 | lsmlub 19696 | . . . . . . 7 ⊢ ((𝑄 ∈ (SubGrp‘𝑊) ∧ 𝑅 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
43 | 31, 33, 41, 42 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (𝑄 ⊕ 𝑅) ⊆ 𝑈)) |
44 | ssnpss 4115 | . . . . . 6 ⊢ ((𝑄 ⊕ 𝑅) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅)) | |
45 | 43, 44 | biimtrdi 253 | . . . . 5 ⊢ (𝜑 → ((𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) → ¬ 𝑈 ⊊ (𝑄 ⊕ 𝑅))) |
46 | 45 | con2d 134 | . . . 4 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → ¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈))) |
47 | ianor 983 | . . . 4 ⊢ (¬ (𝑄 ⊆ 𝑈 ∧ 𝑅 ⊆ 𝑈) ↔ (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) | |
48 | 46, 47 | imbitrdi 251 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ (𝑄 ⊕ 𝑅) → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈))) |
49 | 15, 48 | mpd 15 | . 2 ⊢ (𝜑 → (¬ 𝑄 ⊆ 𝑈 ∨ ¬ 𝑅 ⊆ 𝑈)) |
50 | 18, 40, 49 | mpjaodan 960 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ⊆ wss 3962 ⊊ wpss 3963 {csn 4630 ‘cfv 6562 (class class class)co 7430 0gc0g 17485 SubGrpcsubg 19150 LSSumclsm 19666 Abelcabl 19813 LModclmod 20874 LSubSpclss 20946 LVecclvec 21118 LSAtomsclsa 38955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-oppg 19376 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-lsatoms 38957 df-lcv 39000 |
This theorem is referenced by: lsatcvat2 39032 |
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